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Theorem isfin5 7925
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin5  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )

Proof of Theorem isfin5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fin5 7915 . . 3  |- FinV  =  {
x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x
) ) }
21eleq2i 2347 . 2  |-  ( A  e. FinV  <-> 
A  e.  { x  |  ( x  =  (/)  \/  x  ~<  (
x  +c  x ) ) } )
3 id 19 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
4 0ex 4150 . . . . 5  |-  (/)  e.  _V
53, 4syl6eqel 2371 . . . 4  |-  ( A  =  (/)  ->  A  e. 
_V )
6 relsdom 6870 . . . . 5  |-  Rel  ~<
76brrelexi 4729 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
85, 7jaoi 368 . . 3  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  A  e.  _V )
9 eqeq1 2289 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
10 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1110, 10oveq12d 5876 . . . . 5  |-  ( x  =  A  ->  (
x  +c  x )  =  ( A  +c  A ) )
1210, 11breq12d 4036 . . . 4  |-  ( x  =  A  ->  (
x  ~<  ( x  +c  x )  <->  A  ~<  ( A  +c  A ) ) )
139, 12orbi12d 690 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/  x  ~<  ( x  +c  x ) )  <->  ( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) ) )
148, 13elab3 2921 . 2  |-  ( A  e.  { x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x ) ) }  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
152, 14bitri 240 1  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   (/)c0 3455   class class class wbr 4023  (class class class)co 5858    ~< csdm 6862    +c ccda 7793  FinVcfin5 7908
This theorem is referenced by:  isfin5-2  8017  fin56  8019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-iota 5219  df-fv 5263  df-ov 5861  df-dom 6865  df-sdom 6866  df-fin5 7915
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